Characterization of branching architecture through "universal" ratios of

4168

Macromolecules 1990,23, 4168-4180

Reviews

Characterization of Branching Architecture through “Universal” Ratios of Polymer Solution Properties Jack F. Douglas* Polymers Division, National Institute of Standards & Technology, Gaithersburg, Maryland 20899

Jacques Roovers Division of Chemistry, National Research Council of Canada, Ottawa, Ontario, Canada K I A OR6

Karl F. Freed James Franck Institute and Department of Chemistry, University of Chicago, Chicago, Illinois 60637 Received October 24, 1989; Revised Manuscript Received February 27, 1990

ABSTRACT: Experimental and Monte Carlo data for the dilute-solution properties of “lightly branched” polymers (stars, combs, rings, ...) are compared with the renormalization group predictions of Douglas and Freed. The comparisons focus on ”universal” dimensionless ratios of the mean dimensions of lightly branched polymers, relative to those of linear polymers having the same molecular weight. Complications associated with hydrodynamic solution properties and with the effect of ternary interactions are briefly discussed. Dimensionless ratios involving the polymer second virial coefficient,Az, are also tabulated and compared with theory. means of characterizing the model polymers is through the I. Introduction measurement of dilute-solution properties. For example, Polymer properties can be significantlymodified through a classical measure of branching is the Zimm-Stocka variation of molecular architecture because of two basic mayer ratio,gg,, of the radius of gyration of the branched influences of the topological constraints of branches and polymer to that of the linear polymer at the same molecular loops: (1)The local average polymer density increases weight (see below). Other commonly employed “branching relative to that in a linear chain.’ (2) Cooperative motion parameters” include the g’ and h parameters, defined (see of the chain as a whole is restricted by the constrained below) as the ratios of the intrinsic viscosity and the segmental motion within branches and loops.2 The first hydrodynamic radius for the branched to the linear effect is particularly important in the study of dilutepolymer species, respectively.lOJ1 Dimensionless virial solution properties, while the second effect presumably properties, such as the penetration function, \k, have been dominates at high concentrations.2 largely neglected in characterizations of branched polymers There has been considerable interest lately in using because an adequate theory of excluded volume is required polymers with well-prescribed branching architectures to to interpret such data and because it is only recently that test molecular theories such as the reptation m 0 d e 1 . ~ ~ ~ precise theoretical predictions6p7have been made for \k of Interest also stems from the practical motivation of branched polymers. Section I11 is devoted to these virtailoring the properties of polymeric materials through the ial ratios. control of molecular t ~ p o l o g y .Interest ~ in chain The present paper compares renormalization group (RG) architectures coincides with recent advances in synthetic calculations of Douglas and Freed6 with an extensive procedures? in the theory of polymer solution properties,6*’ and in the capabilities of Monte Carlo simulation methods.8 compilation of d a t a for t h e radius of gyration, This confluence of motivational factors accounts for the hydrodynamic radius, intrinsic viscosity, and second virnumerous recent papers devoted to the influence of id coefficient, in which the branching architecture is varied. branching on polymer properties. Comparison is also made with available Monte Carlo simulation data,12-’6 “scalingtheory”,17and classical meanAn experimental study of how chain architecture affects field calculations.10 Section I1 reviews classical Gaussolution properties first requires the synthesis and precise sian chain predictions for a uniform star polymer and characterization of t h e model polymer species. relates them t o RG computations and Monte Carlo Characterization, in turn, necessitates an adequate theory simulations. The corresponding results for a chain with for interpreting experimental data used to assess the “quality”of the synthesized material. The most important excluded volume6 are summarized, and the relative 0024-9297/90/2223-4168$02.50/0

0 1990 American Chemical Society

Macromolecules, Vol. 23, No. 18, 1990

Characterization of Branching Architecture 4169

Table I Polystyrene (Good Solvent) Radii of Gyration i n Toluene 3-armd 4-arme

6-armf 12-armg 18-armg H-combh

ringi

sample

M x 10-5

(S2) x 1012 cm2

( S2)tina

S141A S181A S191A A-48.2 H5041A H5051A H5121A l2PSlBlA 18PS1 HlAl H3AlA H5A1 HGAlA R19D

5.37 10.61 14.44 18.7 5.2 11.1 18.9 58.0 93.0 4.82 6.82 10.5 17.2 3.34

4.7 11.34 16.78 21.0 3.86 8.65 17.5 33.0 48.0 5.0 7.4 13.0 23.7 2.5

8.4 18.63 26.7 36.1 8.09 19.64 36.6 135.9 236.2 7.4 11.1 18.4 32.8 4.8

g*,(exptl) 0.79 (0.560) 0.608 0.628 0.581 (0.477) 0.440 0.478 0.243 0.203 0.676 0.666 0.706 0.723 0.519

g*ab

gosc

0.778 0.631

0.778 0.625

0.453

0.444

0.248 0.173 0.720

0.236 0.161 0.712

0.516

0.5

Based on (S2)~i, = 1.66 X 10-18MJ7. Theory developed in refs 6 and 7. Results reviewed in ref 10. Reference 32. e References 22 and 39. f Reference 23. 8 Reference 24. Reference 25. Reference 30d. 0

Table I1 Polybutadiene (Good Solvent) Radii of Gyration in Cyclohexane 4-arm

Ce Be

1.434 1.59 3.75 8.3 13.52 3.11 3.40 3.58 5.41 7.62 19.0

Ae

18-armg

800' 16m 1518 18SSC l8SC 2518 3718 9918

*

8

In cm2. Based on ( SZ)tin = 2.55 Reference 57.

X

(1.98) 2.56 6.75 17.0 33.0 (1.54) (1.87) 1.80 3.01 3.97 11.8

11. Universal Ratios of Polymer "Size"as Measures of Branching A common experimental means of characterizing branching architecture is through the ratio of the squared radius of gyration of a branched polymer, ( s 2 ) B r , to the squared radius of gyration, (S')Lin, of a linear polymer having the same molecular weight. This ratio is defined as

g, = ( S 2 ) , r / ( S 2 ) L i n

(2.1)

For a uniform star having f Gaussian chain arms (Le., having arms of equal length) g, is given by the classical Zimm-Stockmayer prediction9 where the superscript

O

(0.616) 0.705 0.674 0.663 0.723 (0.192) (0.21) 0.19 0.195 0.171 0.173

0.631

0.675

0.173

0.161

10-l8MJ% Theory developed in refs 6 and 7. Reviewed in ref 10. e Reference 46. f Reference 56.

insensitivity of such dimensionless ratios to excluded volume is explained by using the RG theory. We also discuss the qualitative influence of ternary interactions on the polymer mean dimensions under 9 conditions.'* Because branching increases the local polymer segmental density, the probability of ternary interactions is increased. Thus, ternary interactions can give rise to a significant contribution to the dimensions of a branched polymer at the 8 point.& Experimental and Monte Carlo data, summarized below, exhibit deviations from the ideal Gaussian chain theory for many-arm stars, which support theoretical expectations of the influence of ternary interactions.18 Dimensionless ratios of hydrodynamic properties, while relatively simple to obtain experimentally, present a whole host of problems in their i n t e r p r e t a t i ~ n . ' ~This ? ~ ~ issue is addressed briefly in section 11.

go, = (3f

3.21 3.63 10.01 25.6 45.7 8.03 8.92 9.48 15.45 23.17 68.28

- 2)/f

(2.2)

indicates a Gaussian chain

property. Equation 2.2, of course, applies only to the limit of long flexible chains. Variations of go, with branching architecture are reviewed by Yamakawa'o and Bywater." The experimental problems of incomplete coupling of the star arms to their central vertex and the role of polydispersity of the star arms on go, are discussed by Bywater." Mansfield and Stockmayer21 generalize ( 2 . 2 ) to the wormlike star model, which is relevant to stars having relatively short or stiff arms. A. g,in Good Solvents. It has long been observed that the g, ratios for experimental good solvent systems closely coincide with the predictions of Gaussian chain statistics.ll Further amplification of this observation is provided by the RG calculations by Douglas and Freed6 and by Monte Carlo simulations.15J6 Since recent theoretical studies have reiterated this point, we simply emphasize the experimental data in support of the theoretical predictions and briefly indicate a theoretical explanation of the insensitivity of g, to excluded volume. Table I summarizes experimental values of g, for 3-, 4-, 6-, 12-, and 18-arm stars, H-combs, and ring polymers for polystyrene in the good solvent toluene. Table I1 considers 4-and 18-arm polybutadiene stars in the good solvent cyclohexane. A superscript asterisk denotes the good solvent limit (e.g., g*,). The tables also provide both the renormalization group (RG) calculations of g*, by Douglas and Freed6 and the classical Gaussian chain computations for go,. Agreement between theory and experimentB%is quite reasonable, although it is difficult to conclude anything more quantitative than the approximation g*, = gosgiven the small difference between the theoretical estimates of g*, and go,. The RG prediction that g*, = go, is certainly in good accord with the experimental observations, and the magnitudes of the experimental values of g, in a good


4170 Douglas et al.

-

arbitrary static radial polymer property Q, scaling as a radius to the pth power (Q R p ) , has a universal scaling function as a function of excluded volume

Table 111 c. Lattice Monte Carlo Data for H-Combs theory

I _

lattice type fcc bcc cubic tetrahedron

simulation: g*,(SAW)a 0.71 f 0.02 0.72 f 0.02 0.71 f 0.02 0.70 f 0.03

g*,(RG)b 0.720

(go,)‘

0.712

*

a Reference 15b. Theory developed in refs 6 and 7. Theory reviewed in ref 10.

solvent generally agree with the average experimental values cited previously by Douglas and Freed.6 Next it is useful to compare Monte Carlo estimates12-16 of g*, and the corresponding RG predictions. Pertinent Monte Carlo data for self-avoiding uniform star polymers have recently been summarized by Barrett and Tremain.16 Figure 4 of ref 16 compares the RG predictions of Douglas and Freed‘Q7 with simulation data for stars with up to 18arms. The agreement between theory and the Monte Carlo data is quite good over the entire range off. Table I11 also lists the H-comb simulations of Lipson et al.15 for g*, on various lattices, where again similar agreement is obtained with RG theory. It should be appreciated that for many-arm stars the perturbation of the excluded-volume interaction, relative to a linear chain, becomes large so that the renormalization group theory no longer provides a useful description. Application of the RG theory should be restricted to f < 12, and only senseless results can be expected from the theory in the infinite-arm limit. The large f limit needs to be treated by separate methods such as scaling theory,17 mean-field theory,10>28 or 1/ f perturbation theory.6e No method can be expected to apply over the entire f range. We briefly mention relevant results of the scaling theory and a mean-field approach, discussed in Appendix A, which complement the RG predictions summarized in Table I. There have been several recent attempts to test the Daoud and Cotton17“scaling theory” prediction, g*, f W 5 for large f , where the prefactor is unspecified. However, it is difficult t o discern from t h e Monte Carlo or experimental data a small variation of the exponent o f f from its classical Gaussian value of -1. Indeed, Whittington et al.15 find an equally good fit to Monte Carlo data for g*, with either the -1 or -4/5 exponents by just adjusting the prefactor. The usefulness of the scaling theory is evidently limited by the uncertainty in the prefactor. An argument is presented in Appendix A, based on the classical modified Flory theory, which gives the Daoud-Cotton scaling result17for g*, along w i t h t h e p r e f a c t o r as

-

g*B(modifiedFlory)

- 1.94f

-4/5

f

+

00

(2.3)

Equation 2.3 is in reasonably close agreement with the Monte Carlo estimates of Whittington et al.l5 and of Barrett and Tremain.16 The prefactor is deduced to be about 1.8-1.9 from the simulation data (see Appendix A). DiMarzio and Guttman29 have also introduced a meanfield theory, augmented by geometrical arguments, that recovers the scaling result g*, f -4/5 along with an estimate of the prefactor. Experimental data,22-26Monte Carlo and direct enumeration studies,12-’6 and RG calculations by Douglas and Freed6vZ7all point to the simple observation that g*, is remarkably close to Zimm and Stockmayer’sestimate for Gaussian chains. The reason for this insensitivity of g, to excluded volume (above the 0 point) is quite simple in the RG theory. Douglas and Freeds show that an

-

Q = GQ(S2)op/2 PQ(z2) (2.4) is the radius of gyration of a Gaussian chain, Here ( S2)op/2 and GQis a number (prefactor) obtained by calculating Q in the Gaussian chain limit, where the dimensionless excluded-volume parameter z2 vanishes and where by definition Ep(z2 = 0) = 1and P (z2 = 0) = 1. The reader is referred to Douglas and Freed2 and Freed7 for a detailed discussion of the specific analytic form of (2.4) and its theoretical derivation. The point to be emphasized here is the “exponent function”, E&), contains most of the excluded-volume dependence and depends only on z2 and the power p of the radial property. E,,(zz)does not depend on the type of radial property (aside from the index p ) or on the chain architecture. The “prefactor function”,P Q ( z ~ ) , on the other hand, depends on the details of the particular measure of the mean dimensions [e.g., (R2),(S2),...I and on the branching architecture (linear, star, comb, ...) and varies only w e a k l y with excluded volume. Further, by forming a dimensionlessratio between measures of polymer size for a branched and a linear polymer a t the same molecular weight, we obtain Q(branched)/Q(linear) = [PQ(z2,branched)/ PQ(z2,linear)]goQ (2.5a) gQ Q(branched)/Q(linear); gQ(z2= 0) = goQ

(2.5b)

so the predominant contribution t o t h e variation of Q with excluded v o l u m e (i.e,, the EP(z2)factor) exactly cancels in f o r m i n g t h e r a t i o gQ. This canceling contribution basically reflects the uniform expansion of the polymer dimensions with excluded volume. T h e ratio PQ(zz,branched)/PQ(zz,linear) represents the relative variation of PQ,due to variation of branching architecture, and is even more slowly varying than is either Pg(z2,linear)or PQ(z2,branched) alone (see Douglas and Freed6 for details). Consequently, the observed insensitivity of g, to excluded volume in experiments and Monte Carlo simulations is perfectly understandable on the basis of RG theory. This insensitivity is predicted t o apply to a wide class of dimensionless ratios of the type gQ. B. 8 Condition Ratios. The measurement of g, for 0 point polymers is complicated by the possibility that the 0 conditions for a branched polymer and a linear polymer are not generally the same. Shifts of the 0 temperature between branched and linear polymers have been observed in numerous physical systemsnIm and also in Monte Carlo ~ i m u l a t i o n .A~ decrease ~ of g, with additional branching means that the average chain density within a polymer coil is increased and consequently the probability of ternary interactions (i.e., the simultaneous volume interference of three parts of the chain in a single region of space) is greater in the branched chain than in a linear chain of the same molecular weight. The idealized two-parameter model neglects the role of ternary interactions, and when these effects are incorporated into the three-parameter model of polymer excluded volume, a 0 point shift with branching is predicted.18 0 point shifts arising from a variation in branching architecture were calculated long ago by Candau et al.3O using a three-parameter Flory mean-field theory. Recent mean-field calculations by DiMarzio and Guttmana also indicate a 8 point shift with branching architecture. Theoretical predictions based on the three-parameter excluded-volumemodel indicate that the ratio of the mean dimensions of the 0 star polymer (measured by the average


Characterization of Branching Architecture 4171 Table VI gsOff-Lattice Monte Carlo Data

IV e,ExperimentalTable Data for Star Polymers

f

P,(exWa

3 4 6 8 12 18

0.82 0.63 0.46

H-comb

0.70

0.28-0.41

P,(exptUb 0.65, 0.63 0.46 0.276 0.228

P,(exptl)‘ 0.65 0.46 0.415 0.33

0.778 0.625 0.444 0.344 0.236 0.161 0.712

Reviewed in ref 32. Polystyrene in cyclohexane 34.5 “C. Polystyrene in cyclohexane 34.5 “C. Reviewed in ref 42. Polyisoprene in dioxane 32.5 “C. In ref 33 @, = 0.42, 0.35, and 0.28 for 8-, 12-, and 18-arm stars, respectively. Theory reviewed in ref 10. Deviations from the Gaussian chain theory are expected as a consequence of ternary interactions (see text). a

* Reviewed in ref 42.

Table V gsLattice Monte Carlo Data under 0 Conditions f

$e

3 4 6 8

-0.52 4.50 -0.48 -0.47

simulation P,(lattice)a P,(lattice)b 0.81 0.66 0.49 0.40

0.83 0.68 0.51 0.42

theory pl(approx)c go$ 0.79 0.65 0.49 0.42

simulations

goad

0.778 0.625 0.444 0.344

a Lattice predictions for P, of Kolinski and Sikorskial on a diamond lattice at cited 8 conditions. Data obtained from Figure 3 of ref 31. 6 is an attractive nearest-neighbor interaction. Results of Mazur and McCrackin12 for a simple cubic lattice where the 8 conditions are taken to be that of the linear chain, = 0.275. These authors indicate that a small variation of with chain architecture is possible but such variations are less than their accuracy in determining +e. Estimated roughly on the basis of the three-parameter theory. See discussion in text and ref 18. Theory reviewed in ref 10.

star arm length) to the mean dimensions of the 8 linear polymer is greater than predicted by the Gaussian chain theory and that the deviations increase with the number of chain arms.18 We next provide evidence that Monte Carlo and real experimental data for g, under 8 conditions (geJ tend to give larger values of ges than the idealized Gaussian chain predictions. Observed deviations grow with increased branching12J4 as predicted by the threeparameter theory.le Table IV lists experimental data forges of star polymers over a range of f.32v33 Note the increased deviation of go, from go, as mentioned above and as emphasized by Huber et al.34 The experimental values of ge, tend to be dispersed for larger f , suggesting that these ratios may no longer be universal numbers. Difficulties in synthesizing “perfect”many-arm stars should be appreciated, however. Observe that deviations from the Zimm-Stockmayer relation (2.2) tend to occur for f > 6 (see Bywaterll). Further, Table V presents ges from Monte Carlo data of Kolinski and Sikorski,3l where the g, values are determined at the values of the lattice polymer-polymer interaction parameter q5e that these authors identify with the 8 conditions of the star.% Mazur and McCrackin12previously found that the magnitude of ges is greater than the predictions for Gaussian chains-about 6 76 greater for three-arm stars and as much as 15% greater for ninearm stars. Off-lattice Monte Carlo simulations by Freire et al.“ display a similar trend forge,, and data from this work are presented in Table VI. Tables V and VI show that the deviation between the lattice model predictions for ge8 and go, increases with the number of chain arms. Mazur and McCrackinl2 and Kolinski and Sikorski3l ascribe the discrepancy between lattice results for ge, and the Gaussian chain predictions of Zimm and Stockmayerg to an inadequacy of the ideal Gaussian chain model to describe a polymer under 8 conditions.

f ~~

theory

Pea

P*P

poab

0.51 f 0.01 0.35 f 0.01 0.28 f 0.01

0.49 f 0.03 0.33 f 0.02 0.21 f 0.01

0.444 0.236 0.160

p*.(RG)c~d

~~

6 12

18

0.453 0.248 0.173

a Reference 14. The star arms are relatively short in these simulations. Results summarized in refs 9 and 10. Theory developed in refs 6 and 7. Observe that the RG theory predictsg*,/ go, > 1while the Monte Carlo data indicates that the corresponding ratio is less than unity for f 2 6.

Calculations for the radius of gyration, ( S2),for stars with ternary interactions have not yet been performed, and this technical calculation is not pursued here. However, a rough estimate of the expansion of g, under 8 conditions of the star can be obtained from available calculations of the expansion of an arm within a star relative to a linear chain of the same molecular weight under 8 conditions.’e Assuming for the sake of rough argument that ( S 2 )for the star expands to the same degree as an individual arm within the star (a uniform expansion-type approximation), neglecting the difference between the 8 condition of the star and linear polymer, and taking the ternary interaction parameter 23’ on the order 23’ 0.01 (consistent with magnitudes deduced from an analysis of lattice models by Cherayil et a1.9, we obtain the approximate predictions for 8,given in Table V. The rough three-parameter theory estimates certainly exhibit the correct trend in the data: the estimates yield reasonable magnitudes for the deviation between ge, and go, and an increasing deviation between ges and g o , as t h e number of arms is increased. Consequently, we entirely agree with Mazur and McCrackin12that the deviation of ge, from go, is due to an inadequacy of the noninteracting Gaussian chain model for 8 chains. However, it should be possible to correct this theoretical deficiency by incorporating ternary interactions in RG calculations for star polymers. According to the RG predictions of the previous subsection, the g, ratio should vary weakly with excluded volume and should increase slightly upon going from a 8 (“ideal”)solution to a good solvent. This prediction is in accord with Monte Carlo simulations of g, for idealized random walks and self-avoiding walks (see Figure 4 of ref 16). Monte Carlo calculations by Mazur and McCrackin12 on a lattice with a nearest-neighbor attractive interaction, however, yield a rather different variation: g, decreases upon going from a 8 to a good solvent, where the “ideal” value of g, is only approached in the good solvent limit. This decrease of g, upon going from 8 to good solvents is also found in the off-lattice simulations of Freire et al. using a 6-12 potential14 (see Table VI). The difference between ge, and go, cannot be attributed to just the difficulty in locating the 8 point of a star since ges tends to be always greater than go, (see Figure 6 of Mazur and McCrackin12). T h e wormlike star calculations of Mansfield and StockmayerZ1indicate that g, (worm star) is less than go,, so the effect cannot be explained as due to chain stiffness in the star arms. The qualitative trend in Mazur and McCrackin’s data for g, is quite understandable on the basis of the threeparameter modell8 and is worth discussing since it underscores theoretical developments necessary to accurately describe the configurational properties of stars and other lightly branched polymers. As discussed above, the three-parameter theory indicates that the mean 8point dimensions of a star polymer relative to those of a linear polymer should generally be larger than predicted

4172 Douglas et al.


by Gaussian chain statistics; however, expansion of the least comment on some of the complicating factors that chain decreases the probability of ternary interactions and should be k e p t in mind i f t h e s e hydrodynamic thus decreases the influence of ternary interactions on the measurements are used t o characterize branching architecture. mean dimensions of branched polymers. It is easy to show from scaling considerations (Appendix B) that ternary It has long been recognized that the hydrodynamic gQ interactions for long swollen chains are "irrelevant" in three factors for star polymers, calculated from the preaverdimensions. However, the RG theory has not yet aged Kirkwood-Riseman and Rouse-Zimm theories, do developed to a point that allows the description of binary not agree very well with the experimental data and that and ternary interactions over the full range of both the deviation increases with the number of star arms." On interactions, and the specific manner in which the ternary the other hand, the claimed agreement for rings between interactions diminish upon chain expansion cannot yet be preaveraged hydrodynamic theories and experiment has described quantitatively. Recently, Douglas and K ~ s m a s ~ ~been reasonably good,6338a result which can only reflect have proposed a promising analytic scheme to describe a a large cancellation of errors in these hydrodynamic gQ related type of multiple interaction problem and have ratios.lg applied it to the description of surface interacting polymers Recent Monte Carlo calculations by Zimm13 and by with excluded volume. Expansion of a chain at a surface Freire et a1.14849 and analytical dynamical RG calculations18 decreases the probability of surface contacts and diminishes stress the importance of the preaveraging approximation, the influence of the surface interaction in much the same which can affect the absolute magnitudes of t h e fashion as expanding a chain in solution diminishes the hydrodynamic radius and the intrinsic viscosity of linear role of ternary interactions. chains by roughly 10% in the asymptotic "nondraining" Although it is difficult, at present, to quantitatively (strong hydrodynamic interaction) limit. Monte Carlo describe the observed decrease of g, upon chain expansion, simulations also show that the effect of the preaveraging we can understand this basic trend as arising from the approximation depends on the chain architecture, so that decreasing influence of ternary interactions on the mean preaveraging errors are not entirely eliminated by taking dimensions combined with the very small increase of ratio ratios of hydrodynamic properties. Near cancellation of g, coming from binary excluded-volume interactions. Mapreaveraging errors may occur in rings, explaining the zur and McCrackin's datal2 underline the necessity of apparent success of the preaveraging theory in this case. developing a version of the RG theory that allows a This suggestion remains to be checked by theoretical consideration of polymer properties over a large range of cal~ulation.3~ both binary and ternary interactions. Another factor to be considered is the presence of ternary C. H y d r o d y n a m i c P r o p e r t i e s . T h e simplest excluded-volume interactions that can cause the gQfactors experimental means of characterizing branching to deviate from the idealized Gaussian chain theory as architecture is through measurements of the intrinsic discussed in the previous section. Calculations by Cherviscosity and the diffusion coefficient (or equivalently the ayil et a1.18show that hydrodynamic properties such as RH hydrodynamic radius). This situation is natural given the are especially sensitive to ternary interactions. relative complexity of measuring (S2)by light or neutron Finally, hydrodynamic properties have the troubling scattering. As in the case of the radius of gyration, it is complication of the "draining effect" (finite hydrodynamic traditional to form ratios of the intrinsic viscosity, [VI, and interaction).20 Expansion of a polymer chain in a good of the hydrodynamic radius, RH, of the branched species solvent lowers the chain density inside the coil and may to the corresponding linear polymer properties, where the thereby diminish the extent of hydrodynamic interaction, comparison is generally taken such that the branched and thus reducing the effective "hydrodynamic size" of the linear species have the same molecular weight.1° Here we polymer relative to the "nondraining limit" (Le., strong denot,e these ratios as hydrodynamic interaction). If this partial draining effect occurs to a different degree in the branched and linear (2.6) species, the hydrodynamic gQ ratios could potentially (2.7) display a sensitivity to the extent of hydrodynamic interaction in good solvents, and this would mean that the The usual convention is to denote g, by g' and gH by h, hydrodynamic gQfactors would depend somewhat on the but this notation is avoided because of the proliferation particular polymersolvent system (an effect not considered of g ratios for the many properties of interest and because in current Monte Carlo simulations). Complications from h is already utilized conventionally for the dimensionless partial draining should be less problematic under 0 hydrodynamic interaction parameter. Barrett and conditions where neglecting the finite magnitude of the Tremain16utilize g' to denote the expansion of an arm in hydrodynamic interaction strength should be a better a star polymer relative to a linear chain, and this definition approximation.10p20 unfortunately overlaps with the traditional hydrodynamic theory notation. The gQ notation with a subscript Q, If the extent of hydrodynamic interaction is not affected indicating specifically what property is being considered significantly by t h e branching architecture ( t h e in the ratio, is strongly suggested as a standardized notation dimensionless hydrodynamic interaction parameter in the (see ref 37). The ratio gQis given a 0 or * superscript to continuum polymer hydrodynamics theory is not affected denote t h e solvent condition, 8 o r good solvent, by branching), then gH and g, are expected to vary slowly respectively. with excluded-volume and hydrodynamic interactions. Unfortunately, the relative low cost and simplicity of Thus, in spite of the theoretical difficulties of including the measurement of hydrodynamic properties are preaveraging corrections, excluded volume, and a somewhat counterbalanced by the complexity and relative hydrodynamic interaction dependent on excluded volume,1g inadequacy of the corresponding theory of polymer we can still expect the experimental gH and g, ratios to hydrodynamics. The difficulties are, indeed, so severe that be insensitive to excluded volume and to have values we discuss the theoretical and experimental situations with s i m i l a r t o t h o s e e s t i m a t e d from M o n t e C a r l o some trepidation. But given the prevalence of estimating ~imulationl3J~,~g by Zimm and others without employing various hydrodynamic gQratios we feel compelled to at preaveraging. This expectation is supported by an


Macromolecules, Vol. 23, No. 18, 1990 Table VI1 Variation of & with Chain Architecture experiment polymer PS" PIb PSc PId PSc PSr

4-arm star 6-arm star 8-arm star 12-arm star

18-arm star

PId PSg PId PBdh

ring

Psi

PH

eqs 2.8 and 2.9

~FH

g*H

0.94

0.93

0.89 0.85

0.86 0.81

0.79

0.73

0.73

0.63

g*H

0.93 0.92 0.86 0.82 0.70 0.830.73 0.77 0.68 0.64 0.68 0.835

0.94 0.89 0.81 0.850.74 0.76 0.72 0.890.86

0.858

Pi9

Polystyrene: 8 solvent, cyclohexane (35 "C); good solvent, toluene (35 "C). Reference 39. b Polyisoprene: 8 solvent, dioxane (34 "C); good solvent, toluene (35 "C). Reference 40. Reference 41. Reference 40. e One high molecular weight sample. Reference 24. f gH decreasing with increasing molecular weight. This set of polymers includes some stars with very small branches. Reference 34.8 Reference 24. h Polybutadiene: 8 solvent, dioxane (26 "C); good solvent, cyclohexane (25 "C). Reference 42. gH decreasing with increasing molecular weight, 1.8 X l o 4 < M , < 4.3 X lo5. Reference 26. j Reference 43. Table VI11 Variation of a*with Chain Architecture 0

~~

eqs 2.8 and 2.9 experiment polymer 3-arm star &arm star 6-arm star 8-arm star 12-arm star

PS" PSb PI' PBdd PSe PIf PI8

PSh Psi PSS

18-arm star

PSh PIS PBdj PBdk

H-comb ring

PS' PSm PS"

PSO PBdp

gen

0.856 0.76 0.772 0.760 0.63 0.625 0.531 0.41 0.42 0.422 0.35 0.298 0.284

g*,

0.833 0.724 0.73 0.73 0.57 0.589 0.43 0.35 0.345 0.328 0.26 0.242 0.225 0.26 0.80 0.73 0.66- 0.670.60 0.56 0.71- 0.790.63 0.68 0.71- 0.760.66 0.56 0.63

6:

g*,: (empirical) (empirical) 0.86 0.85 0.76 0.73 0.63

0.57

0.54 0.43

0.47 0.34

0.35

0.23

0 Polystyrene: 8 solvent, cyclohexane (35 "C); good solvent, toluene (35 "C). Reference 32. b Reference 39. Polyisoprene: 8 solvent, dioxane (34 O C ) ; good solvent, toluene (35 "C). Reference 5a. d Polybutadiene: 8 solvent, dioxane (26 "C);good solvent, toluene (35 "C). References 45 and 46. Reference 23. f Reference 5a. 8 Reference 40. h One high molecular weight sample. Reference 24. Reference 32. J Reference 57. Polybutadiene: good solvent, cyclohexane (25 "C).Reference 42. Reference 25. Decreases with increasing molecular weight. Reference 26. Reference 3b. Decreases with increasing molecular weight M,, < 30 OOO in a 8 solvent of 0.67-0.66 except a t very low molecular weight M,,== 3000. P Reference 48.

approximate hydrodynamic theory under the assumption that the hydrodynamic interaction is independent of branching structure.20 Experimental values of the hydrodynamic gQratios are commonly found to be nearly independent of solvent to within experimental precision and to be in relatively good agreement with Monte Carlo simulations,M when account of the preaveraging effect is made. We now proceed to discuss the experimental data,

Table IX Monte Carlo Data for Hydrodynamic Parameters of Star-Branched Polymers 3

4 6 12 18

0.97O 0.8gC 0.82c 0.73c

0.94"*b 0.89"~~ 0.7gC 0.68'

0.54' 0.39 0.28c

0.85" 0.73Osb 0.57,O 0.56b 0.37c 0.22c

0 Reference 50. Reference 13. References 49 and 14. Simulation values of gH and g, for Gaussian chains (see Table 111 of ref 49) are smaller than values of PHand p, for 8 point polymers where the discrepancy increases with the number of arms. This effect is likely due to ternary interactions. Simulation data for chains without excluded volume (Le., Gaussian chains) should be carefully distinguished from 8 point chain data.

keeping in mind the limitations of the hydrodynamic theory. Tables VI1 and VI11 summarizegH and g, data for regular stars, rings, and H - ~ o m b s . ~ ~The ~ ~data ~ *in~these ~ ~ ~ * ~ ~ tables are obtained by first fitting the linear chain properties Q to a scaling equation Q 0: Ma with M the molecular weight. Properties of linear polymers are calculated from these scaling equations at the molecular weights for which the branched polymer data are available, and the results are combined with the branched polymer data to obtain the cited dimensionless ratios for the viscosity and the hydrodynamic radius. These ratios are then averaged over sets of data for a particular type of branching architecture. Recent Monte Carlo estimate^^^^'^^^^^^ of g, and gH have been made without introducing the preaveraging approximation but retaining the rigid-body approximation of the standard Kirkwood-Riseman theory. A summary of recent results is given in Table IX. These Monte Carlo calculations apply to the nondraining limit only. Simulation methods describing variable hydrodynamic interaction have not yet been devised. Experimental values of g, and gH in 0 and good solvents are evidently consistent with simulation estimates. Observe that the ratios g, and gH generally decrease with increasing solvent quality and that this trend is reproduced by the Monte Carlo simulation data. In contrast, the gQratios for static properties and light branching (small f ) tend to be increasing functions of excluded volume. The tendency of g, and gH to increase with solvent quality is not yet understood, but it seems likely that a decrease of ternary interaction with swelling is responsible for part of the trend. As mentioned before, analytical computations of g, and gH using the preaveraging approximation tend to be rather inaccurate. Roovers and co-workers51have noted useful correlations between ge, and geH and the number of star arms f as ge,(empirical) = [ ( 3 f - 2)/f 2]0.58

(2.8a)

These empirical forms were motivated by results from the preaveraging theory. Further, a semiempirical correlation of the variation of g, and gH with excluded volume is described by Douglas and Freed.6 The g, and gH ratios in good solvents (f I2) are given by6 g*,(empirical) = ge,[l - 0.276 - O.O15(f- 1)]/(1 - 0.276) (2.9a)

4174 Douglas et al.

Macromolecules, Vol. 23, No. 18, 1990 Table X 4 ' Linear Polystyrene in Toluene at 35 O C

g*H(empirical) = geH[l - 0.068 - 0.0075(f- 1 ) ] / ( 1 - 0.068) (2.9b)

and values of g, and gH are listed for comparison in Tables VI1 and VIII. Equation 2.8 is used to estimate geqand geH in (2.9). There is a remarkable consistency between the experimental and Monte Carlo data and the simple empirical correlations for g, and gH in (2.8) and (2.9). In a previous paper we show that the excluded-volume dependence of hydrodynamic properties ([VI, RH) of different polymers can display a significant variation5* attributed to the "draining effect". The dimensionless ratios g, and gH, however, are rather insensitbe to excluded volume and seem to be insensitive to the particular polymer-solvent system considered. Superficially the experimental data seem to be consistent with the nondraining nonpreaveraged Monte Carlo calculations, even though the hydrodynamic properties of linear polystyrene do n o t conform very well with t h e nondraining hydrodynamic theory. The universality exhibited by g, and gH seems to be the consequence of a fortuitous cancellation of ternary, hydrodynamic, and excludedvolume effects in these dimensionless ratios. Further experimental work on a variety of polymersolvent systems is needed to establish whether the apparent universality of g, and gH holds generally. 111. Virial Ratios and the Penetration Function,

4 The polymer second virial coefficient, A2, is another property that is sensitive to branching architecture. Until recently, however, the theory of polymer solutions could provide only a rather limited description of A2, and as a consequence this parameter has been neglected relative to the Zimm-Stockmayer ratio, g,. Renormalization group calculations by Douglas and Freed6 give precise predictions for the variation of A2 with chain branching and excluded volume. A. Linear Polymers. Before considering how branching architecture affects A2, it is useful to review theoretical predictions for the linear polymer and to compare these results with available experimental data. The RG theory is, after all, a perturbative method, and this kind of testing is essential in establishing the predictive capability of the theory. Usually, a dimensionless measure of A2 is used in presenting data in the experimental literature, and the conventional measure is the penetration function, @, defined in three dimensions as6,7J0 @ = ( 2 A @ / N A )/ ( 4 a ( S2))3/2

(3.1)

where M is the polymer molecular weight and N A is Avogadro's number. Renormalization group theory predicts that the penetration function for long monodisperse samples of linear flexible polymers in three dimensions is a universal constant in good solvent, which in second order equals6 @* = 0.269 (3.2) Further, @* is computed as somewhat smaller for polydisperse samples, marginal solvents, and only modestly long chains.53 Universal scaling curves describing the crossover of @ from its value of zero under 0 conditions to its limiting value of @* in good solvents are given by Douglas and

Table X provides experimental data5bv41 for \k* of linear polystyrene in the good solvent toluene at 35 "C along with the associated thermodynamic properties, sample

O* (good

sample

C6bhb PS800f2C C7bbb PTd PS1000f2C SSIAd

M,

X

6.65 6.73 8.60 15.3 17.0 68.0

A2 X lo4 a

( S 2 )X 10l2

solvent)

3.40 2.81

4.18 10.68 10.81 14.8 21.5 35.3 170

0.239 0.265 0.265 0.255 0.282 0.225 0.244 0.269

2.19

2.62 2.33 2.19 1.57

RG predictione

I n mol.cm3/g2; log A 2 , ~ i "= -2.15 - 0.242 log M,. 10-1*M,1.17cm2 quoted in ref 5b, Table I. Reference 41. Roovers, unpublished data. e Theory developed in refs 6 and 7. a

( S 2 ) ~= i ,1.66 X

Table XI 4*Linear Polybutadienes in Cyclohexane at 25 OC

M, X samplea PBD 120 1.14 165 1.55 170 1.64 180 1.80 300 2.56 400 3.61 800 7.60 RG predictiond

A2 X lo4

*

10.5 10.0 9.76 9.35 8.69 8.27 6.85

( S 2 )X 10l2

2.31 3.50 3.73 4.13 6.64 9.40 22.9

**(good solvent) 0.290 0.274 0.272 0.26s 0.248 0.258 0.269 0.269

Samples described in ref 54. * log A2 = -1.880 - 0.2128 log M,; in mol.cm3/g2. (s2) = 0.0255W1m. Theory developed in refs 6 and 7.

identifications, and references. A similar tabulation of data for polybutadiene in the good solvent cyclohexane" is given in Table XI. Agreement between theory and experiment seems to be reasonably good, and the range of values for \k* is consistent with the earlier estimate of Douglas and Freed6 given by @*(empirical)= 0.26 f 0.04 (3.3) A precise Monte Carlo simulation of @* would be welcome. B. Variation of the Virial Coefficient with Branching Architecture. There are two particularly convenient measures of A2 that are useful in characterizing branching architecture. First, there is the analogue of the g, ratio defined by the ratio of A2 for the branched and linear polymer as6 gA = A2,Br/A2,Lin (3.4) Renormalization group calculation^^^^ indicate that g*A is a constant in a good solvent for a given branching architecture. Table XI1 presents22g23~32v39 the ratio g*A for three-, four-, and six-arm regular stars of polystyrene in toluene at 35 "C. The RG predictions for these ratios are g*A(f=3) = 0.968, g * A ( f = 4 ) = 0.923,

g*,(f=6) = 0.808 (3.5) which are to be compared with the experimental values in Table XII. Table XI11 lists empirical e s t i m a t e s ~ - ~ ~ ~ * ~ of &?*Afor H-combs, rings, and some many-arm stars, where the theoretical estimates of g*A(H-comb) = 0.949 and g*A(ring) = 0.879 agree reasonably well with the data. It should be borne in mind that the theoretical calculations for rings apply t o a kind of "ideal ring" where an equilibrium ensemble of all types of possible knotted structures are implicitly involved in the statistical a ~ e r a g i n g ."Real ~ ~ rings" do not necessarily correspond to this ideal, and it would be interesting to pursue the question of how specific constraints, corresponding to



Table XI1 9* Regular Star Polystyrenes in Toluene at 35 "C

sample

M,

X

A2X

A*2gr/ lo4 (s?) X 10l2 A*zJ,~ \k*

sample

Three-Arm Stars" SS-13 SS-14 SS-15 SS-16 RG predictionb

3.92 7.75 9.20 11.9

S141A S181A S191A A-48,2d RG predictionb

5.37 10.61 14.44 18.7

HS041A HS051A HSl2lA RG predictionb

Table XIV Virial Properties of 4-Arm Star Polybutadienes in Cyclohexane at 25 "C

Four-Arm StarsC 2.80 4.7 2.34 11.34 2.04 16.78 2.03 21.0

Six-Arm S t a d 5.20 2.51 3.86 11.1 2.15 8.65 18.9 1.79 17.5

0.968

0.41 0.53 0.40 0.41 0.384

0.963 0.950 0.892 0.945 0.923

0.590 0.516 0.461 0.550 0.517

0.857 0.882 0.835 0.808

0.667 0.776 0.651 0.793

a Reference 32. References 6 and 7. e Reference 39. Reference 22. e Reference 23. f Table IX of Bauer et presents for polyisoprene stars over a large range off. According to this work: (linear) = 0.25,3*(f=8) = 0.77, \k*(f=12) = 1.43; 9*(f=18) = 1.53. The value of 'II*for large f is found to be close to the hard-sphere value **(HS) = 1.62 for f 2 12 (see Appendix C).

** **-

C" B" A" PBD4S800B3' PBD4S1600B3b RG predictionc

Mw X

A2x

LO4

10l2

A*2,~r/ A*2~i

Q*

1.434 1.59 3.75 8.3 13.52

9.51 9.08 7.28 6.40 5.89

(1.98) 2.56 6.75 17.0 33.0

0.960 0.938 0.906 0.947 0.970 0.923

(0.523) 0.418 0.435 0.46s 0.423 0.517

(S2) X

Synthesis given in ref 46. Synthesis given in ref 56. Theory developed in refs 6 and 7. Table XV Virial Thermodynamic Properties of 18-Arm Star Polybutadienes in Cyclohexane at 25 OC sample"

M, X lo6

PBD1518 PBD18SSC PBD18SC PBD2518 PBD3718 PBD9918

3.11 3.40 3.58 5.41 7.62 19.0

A2 X

lo' (s2) X

3.00 3.05 3.18 3.03 2.47 1.79

(1.54) (1.87) 1.80 3.01 3.97 11.8

Samples described in ref 57. Quoted mean is 0.37 for A*2gr/ A*2,~hand 1.20 for in ref 57. e See footnote g of Table XIII.

*

computations of \ k * ~ , from Douglas and Freed6 are presented in Tables X-XIV. Experimental values of \k* for polystyrene stars in Table XI11 toluene at 35 OC are given in Table XII, while those for Virial Properties of Branched Polystyrenes in Toluene at 35 OC H-combs, rings, and many-arm stars are listed in Table XIII. Tables XIV and XV summarize similar but more A*2,~r/ l i m i t e d d a t a 4 6 - 5 6f o r p o l y b u t a d i e n e s t a r s i n sample Mw X A2 X lo4 (S2)X 10l2 A*2,~i" 9* c y ~ l o h e x a n e .Comparison ~ ~ ~ ~ ~ ~ between ~~ the theoretical H-Comb" predictions of Tables X-XV and the experimental values HlAl 4.82 2.95 5.0 0.9% 0.46 shows that the RG theory is rather consistent with the H3AlA 6.82 2.64 7.4 0.963 0.45 H5A1 10.5 2.40 13.0 0.971 0.42 experimental data. Monte Carlo simulations for \k*Br H6AlA 17.2 2.10 23.7 0.958 0.40 would provide a further test of the RG theory but are not RG predictionb 0.949 0.429 yet available. In summary, the second virial coefficient provides a sensitive probe of branching architecture and Rinp R19D 3.34 2.94 2.5 0.90 0 . 6 1 ~ ~ can be described to a reasonable level of accuracy by the RG predictionb 0.879 0.687 renormalization group theory. l2PSlBlA RG predictionb

58.0

18 PS1

93.0

12-Arm StaP 0.83

33.0

18-Arm Stares 0.57 48.0

0.51 0.336

1.35

0.39

1.10

l.lOe

Synthesis and properties are described in ref 25, Table 11. Theory developed in refs 6 and 7. e Reference 3Od. Reference 26. e Properties given in ref 24, Table 11. f Huber et al. quote different values for this sample in ref 55 leading to A * 2 g r / A * 2 ~= 0.50 and = 0.81. 8 The RG prediction for the ratio A * ~ , B ~ / A *becomes ~ , L ~ unphysically negative for f = 18. This result is expected since we are outside the range of applicability of the RG theory (see section IIA). (1

**

different amounts and types of knots, affect configurational properties of ring polymers. The penetration function, \k, provides a second useful measure of the second virial coefficient that is rather sensitive to branching architecture. To first order in the t = 4 - d (d is the dimension) renormalization group ~ ~ computations of \ k * yield6 **B~

= (gOs)-d'2\k*Lin

+ O(t2),

\k*Lin = t/8

+ 0(c2)

(3.6) Good qualitative values of \ k * can ~ ~be obtained by taking \ k * ~ i , , from the second order in the t value \ k * ~ i , , = 0.269 (see Appendix D ) . T h e renormalization group

IV. Conclusion The preparation of model polymers to test structureproperty relationships requires an accurate means to characterize chain architecture. Typically, t h e characterization involves various measures of molecular "size" (the radius of gyration, second virial coefficient, hydrodynamic radius, intrinsic viscosity, ...), which are compared with the corresponding property of a linear polymer of the same polymer type, of the same molecular weight, and under the same solvent conditions. Ideally the set Of gQ ratios of branched to linear polymer properties provide a "fingerprint", specifying uniquely the branching architecture. In practice the characterization of branching using the gg ratios involves a number of subtle effects. For example, the gQ ratios measured under 8 conditions are affected by ternary interactions that introduce an increasing deviation from the ideal Gaussian chain theory with increased branching. This effect is an obvious consequence of greater ternary chain interferences with an increased chain density. Theoretical difficulty in treating such ternary effects is complicated by t h e expected variation of ternary interactions with solvent quality. Hydrodynamic properties have additional complications that limit their usefulness as measures of branching. In addition to a

4176

Douglas et al.


particular sensitivity t o ternary interactions, t h e h y d r o d y n a m i c c a l c u l a t i o n s involve n u m e r o u s approximations, with the preaveragingapproximation being the most notable. No available analytical scheme allows for an accurate estimation of these preaveraging errors, although useful bounds for this effect are provided by Monte Carlo simulation data. Another problem is that hydrodynamic properties depend on a hydrodynamic interaction in addition to the excluded-volume interactions, and this particularly complicates the interpretation of hydrodynamic properties in good solvents because of the possibility of increased draining through expanded polymer coils. All these difficulties are unfortunate given the relative simplicity of performing hydrodynamic property measurements. Scaling arguments (Appendix B) show that ternary interactions are less of a problem in good solvents, and a general discussion using the renormalization group theory indicates that the g*Q ratios should differ rather little from their Gaussian chain counterparts goQ. This situation is rather ironic given the inapplicability of go, to estimate gesbecause of deviations due to ternary interactions. Virial properties are rather sensitive to branching architecture. Hence, good solvent dimensionless ratios, such as the penetration function, are useful parameters for characterizing branching architecture. The virial ratios are especially important in view of the above-cited factors, which complicate the interpretation of the hydrodynamic dimensionless ratios, gQ. Comparison of the renormalization group theory and the virial ratios of carefully prepared model polymers gives encouraging agreement. Sometimes data for the mean dimensions of a polymer are summarized in terms of an "equivalent sphere" notation.4w42There is a tendency for the properties of many-arm stars to approach those of a hard sphere in the limit of large f. It seems that this limiting behavior starts to set in for f 1 20. The equivalent sphere notation is briefly summarized in Appendix C for those who prefer this notation. Discussion of our renormalization group calculations conspicuously omits reference to the calculations of Oono and c o - w o r k e r ~ Reasons .~~ for this omission are described in Appendix D.

Acknowledgment. K.F.F. is partially supported by NSF Grant DMR 86-14358. J.F.D. thanks Greg McKenna a t NIST for his general comments on the paper and for helpful discussions on the characterization of ring polymers. Appendix A. Modified Flory Prediction for pB The modified Flory theory is derived from a synthesis of the classical two-parameter perturbation theory and the Flory mean-field theory.10 Classical Flory theory in d dimensions predicts that the linear chain expansion factor of the end-to-end distance, ( U R = ~ ( R2)/ ( R2)o, is equal to

cRz 2

(AS) where z2 is the Fixman excluded-volume parameter of the two-parameter perturbation theory and Cg is a constant that depends on the spatial dimension d. Stockmayer suggested replacing CR by an expression consistent with the two-parameter perturbation expansion about the z2 = 0 limit (see ref lo), and more recent renormalized perturbation expansion calculations by Edwards and Singh28yield essentially the modified Flory theory as a leading approximation. aRd+2 - a R d

=

Briefly we mention the physical idea behind the modified Flory approximation. Perturbation theory yields the expansion factor a ~ 2 as

(R2)/(R2)o= a R 2 = 1 + CRZ2 + O(ZZ2)

(A.2)

where z2 is given by z 2 = ( d / 2 ~ ) ~ / ~ & " n ~ / ( R ~ )C,(d=3) ; / ~ , = 413

(A.3)

In (A.3) @zO is the binary cluster integral and n is the number of statistical segments. The modified Flory expression is obtained by noting that as the chain expands the volume pervaded by the coil increases as ( R2)odi2 ( R2)d/2.Introducing this replacement into the definition of z2 in (A.3) gives the renormalized measure of the excluded-volume interaction

-

z2

-

= ZZ/aRld

(A.4)

The expression on the right-hand side of (A.4) is denoted as 2 in the classical mean-field theory literature.1° Introducing the correspondence of (A.4) in (A.2) yields the modified Flory result (A.l), which is self-consistent with the perturbation expansion (A.2). At a more rigorous level the Edwards-Singh approach is based on the simple idea of redefining the reference Hamiltonian such that the excluded-volume perturbation remains The method can be summarized in a way that avoids technical complication. We start with an interaction Hamiltonian for chains with excluded volume H / k B T = HO+ H I where HOdesignates the idealized Gaussian chain Hamiltonian and HIdefines the excludedvolume interaction (see Appendix B). We then redefine the Hamiltonian as

H / k B T = aHo + [HI+ (1 - a)HO]

(A.5)

where a is determined by requiring that the contribution to a from the configurational average of HI - (1 - a)Ho vanishes. This again leads to (A.l) with a value of CR slightly different than 4/3.28 Now that we have schematically explained that there is a theoretical and intuitive basis for the modified Flory theory as an approximation scheme, we may apply it to star polymers. The g, ratio for chains with excluded volume is defined by g, = (s2)(star)/(s2)(linear)= gos[as2(star)/

a;(linear)]

(A.6)

=C,d2 asd+2 - a; = Clinearz2, Clinear= 1.276

(A.7a) (A.7b)

where the modified Flory theory gives

ap2- a;

The two-parameter theory expression for C,t, is presented in Yamakawa,'" and for large f we note that'"

- O.430fli2,

f ---* 03 (A.8a) in three dimensions. Combining (A.6), (A.7a), (A.7b), (A.8), and the observation C@ ,

go, = (3f - 2)/f

- 3/f,

f

-

we obtain g, in the good solvent limit as

-

m

-

(A.8b)

g*,(modified Flory) 1.94j"/5, d = 3, f (A.9) Equation A.9 is to be contrasted with the estimates of Tremian and Barrettl6


-

g*,(SAW) 1.86f -4/5, while Whittington et aL15 obtain

Characterization of Branching Architecture 4177 f

+

m

(A.10)

g*,(SAW) 1.83f f (A.ll) The “scaling theory” of Daoud and Cotton’7 is also consistent with (A.9HA.11) but does not provide any specific prediction for the important prefactor. A recent mean-field calculation by DiMarzio and GuttmanB yields the same type of scaling relation as (A.9), except the prefactor is obtained as 1.74. The modified Flory theory can also be used to estimate g, under 8 conditions in the limit f m. Including the ternary interaction 2 3 (see Appendix B) in (A.7) and generalizing the modified Flory theory produces +

-

excluded volume has the asymptotic behavior ( R2) W , where v is greater than its Gaussian value of ‘ / 2 . Introducing the dimensionless units of (B.4) into (B.3) gives

Q,

-

An estimate of Dskrcan be obtained from the expansion factor of an arm in the star. From the perturbation calculation given in Table I1 of ref 18 we expect Dst, p for d = 3 with the constant of proportionality unspecified. Taking 22 = 0 in (A.12) and using the definitions (A.6) and (A.8b) implies

-

(A.13) Interestingly, this is again consistent with the Daoud and Cotton17scaling analysis. The qualitative point we infer from (A.13) is that gesis significantly larger than go, for many-arm stars because of ternary interactions.68 Application of the scaling relation (A.13) is limited, however, because the shift of the star 8 point from the linear chain 8 point cannot be generally neglected in the large f limit.

Appendix B. Irrelevance of Ternary Interactions for Swollen Chains in Three Dimensions

H3 = ( e , / 3 ! ) ~ ’ d r S 0 ’ d x ’ ~ ’ d ~6[f(x) ” - f(x’)] X 6[P(x’) - f ( x ” ) ] ( 2 ~ (B.5a) )~ i 3 ( d / 2 ~ ( R ’ ) ) ~ @ ~ ONn ~ nl

(B.5b)

The ternary interaction in (B.5b) scales with chain length as 13

- n3-2d”

03.6)

Thus, near the 8 point in d = 3 we have v = ‘/zand the exponent in (B.6) is zero, indicating that the ternary interactions may affect numerical prefactors. On the other hand, in good solvents, we have Y > l / 2 and the exponent 3 - 2dv is negative, implying the irrelevance of ternary interactions in swollen chains. (Irrelevance means that the interaction gives no contribution in the limit of long chains.)

Appendix C. “Equivalent Sphere” Notation Experimental data for dilute polymer solution properties are frequently presented in terms of an equivalent sphere notation rather than the dimensionless ratios given in the t e ~ t . ~ ~We p ~briefly + ~ ~explain how to translate from one notation to another. Einstein’s equation for the viscosity of a dilute suspension of hard spheres yields (C.1) where V,,is the equivalent volume of the sphere with radius v q

= 2[v1M/5NA

The continuous chain model represents the chain configuration by a position vector R(T)where T is a contour variable ranging between 0 and N, the polymer chain l e n g t h . A Gaussian c h a i n is described by t h e configurational Hamiltonian in units of kBT as7

(C.2) NAis Avogadro’s number. Similarly, Stokes’ equation provides the hydrodynamic radius RH as

H o / k B T = (d/21)LNldR(~)/drl2 d~

R, = f P V ,

(B.l)

where 1 is the Kuhn length and d is the dimension. The binary interaction of the two-parameter model is represented by the interaction contribution t o the Hamiltonian7

A cutoff, which prevents chain segments from interacting with themselves, is formally neglected in (B.2), and Pz” denotes the usual binary cluster integral. Ternary interactions are similarly represented by the interaction7J8

R,, = [ 3 [ v ] M / l o T N ~ ] ~ / ~

(C.3) where 17, is the solvent viscosity and f is the friction coefficient. Also, hard-spherical molecules have the second virial coefficient relation10 (C.4) where V A is the volume of an “equivalent” hard-sphere molecule. Thus, the hard-sphere virial radius RA is equal (C.5) Finally, we have the static radius defined in terms of the radius of gyration as

where P 3 O is the ternary cluster integral. To understand the effect of chain swelling on H3, dimensionless units are introduced for the position vector, R(T),and the contour length, T

P(x) = R(7)[d/(R2)I1I2 x = T/N

(B.4)

where ( R2) is the mean-square end-to-end distance of the interacting chain. A chain with strong repulsive binary

R, = ( S 2 ) ’ I 2 (C.6) If we equate the physical properties on the right-hand side of (C.2), (C.3), and ((2.5) with the conventionaldefinitions1° of the polymer properties, we obtain the “hard-sphere equivalent radii” (C.7)

4178 Douglas et al.


(C.8)

malized perturbation expansion in the form6v7

+ [7/6 + 4 In 2]u2 + O(u3)

(D.1) where u is the renormalized dimensionless excludedvolume parameter. In the good solvent limit \k* is obtained u*,where6v7 by evaluating (D.l) at the fixed point u \k = u

-

and taking the ratios of these radii gives

u* = (t/8) + ( 2 1 / 4 ) ( ~ / 8+ ) ~O(t3) (D.2) When Oono and Freedm first computed \k*, the fixed point u* was obtained only to order t. Inserting the leading c J8 estimate of u* from (D.2) into (D.l) provides the "one and one half" order approximation (C.lOa) In the unperturbed state this reduces to (PO= 5.99 and 90 = 2 . 5 X a r e used a s e s t i m a t e s w i t h o u t p r e a ~ e r a g i n g l ~ Jfor ~ vlinear ~ ~ chains)

Ro,:Ro,:RoH:RoA= 1:0.836:0.7780 (C.lOb) while for nondraining good solvent^,^^^^ we obtain R*,:R*,,:R*H:R*A= 1:0.805:0.777:0.710 (C.lOC) Bauer et al.40and Roovers and Martin42observe that the static and dynamic properties of many-arm stars approach the limiting behavior of hard spheres for large f . The radius of gyration R, of a uniform density (hyper)sphere of spatial dimension d is related to the sphere radius R by

R: = [ d / ( d + 2)]R2

(C.lla) Equation C.lla leads to the hard-sphere ratios (see ref 42 for a discussion of the d = 3 case)

R,(HS):R,(HS):RH(HS):RA(HS) = l:[d/(d + 2)]'/2:[d/(d + 2)]'/2:[d/(d+ 2)]'/'

(C.llb)

We note that the ratio R,(HS):RA(HS)= [ d / ( d + 2)]1/2 implies that the penetration function \k in the hardsphere limit equals \k(HS) = [ ( d + 2)/3]d/2/I'(l + d/2)

(C.12a)

\k(HS;d=3) = (5/3)3/2(4/3a'/2)= 1.62 (C.12b) where r denotes the gamma function. Equation (2.12 corrects note 36 of ref 6a, which equates the sphere radius R with the radius of gyration R,. For 56-arm polyisoprene stars Bauer et al.40find \k(f=56.2) = 1.65, which is consistent with (C.12b). We also observe that the ratio R,/RH should be rather insensitive to branching architecture. Equation C.10 for linear chains gives the ratios

RO,,IRoH= 1.07, R*,/R*H = 1.03 (nondraining) and for a hard sphere we have by definition R,(HS)/ RH(HS)= 1. Roovers and Toporowski4' observed that R,/ RHis insensitive branching structure (linear, comb, star) and solvent quality; this ratio varies in the range R,,/RH = 1.03 f 0.05. Appendix D. Comparison with Other Renormalization Group Calculations Our discussion of renormalization group calculations of polymer properties has omitted reference to several equilibrium and dynamical computations by Oono and cow o r k e r ~ .This ~ ~ has been done because of technical difficulties in these calculations. Consider first the penetration function, \k, which emerges from the renor-

\k*

= (t/8) + (7/6 + 4 In 2 ) ( ~ / 8 ) ~ ,\k*(d=3) = 0.19

(D.3) that omits the order t2 contribution from (D.2). Using the second-order expression for u* in (D.l)

+ ( 2 1 / 4 ) ( ~ / 8+) ~ (7/6 + 4 In 2)(t/8)' + Ob3),

\k* = (t/8)

\k*(d=3) = 0.27 (D.4) We m e n t i o n t h a t Des CloizeauxG1 calculated a dimensionless virial coefficient, g*, closely related to \k* and consistent with (D.4). The error incurred by the 11/2order expression (D.3) is equal 0.08, which is nonnegligible. Calculations by Oono and c o - w ~ r k e rstill s ~ ~employ the 11/2-order approximation leading to lower estimates of \k* as in (D.3). The problem with the l1/2-order approximation becomes more severe for many-arm stars. Renormalization group calculations by Miyake and FreedGbemploy the l l / z order approximation and lead to values of \k*(f) that have a maximum for f = 4, \k*(f=4) = 0.265, and for larger f the penetration function approaches zero, e.g., \k*(f=15) = 0.0015. The experimental data and our RG predictions,6p7however, indicate a monotonic increase of \k* with f and reasonable values of \k*(f) for f 5 12. However, the RG calculations of Douglas and Freed6J do not predict the approach of \k* to the hard-sphere limit described in Appendix C. The limitations of the continuum model in the large f limit are summarized in section 1I.A. Oono and c o - w ~ r k e rhave s ~ ~ also performed dynamical renormalization group calculations leading to hydrodynamic virial expansions of the form ( D . l ) , corresponding to the hydrodynamic radius, R H ,and intrinsic viscosity, [SI. These calculations neglect the second-ordercontributions to the hydrodynamic fiied point (again the 11/2-order approximation) and lead to spurious conclusions. For example, these calculations indicate no preaveraging correction for the hydrodynamic radius, which is in contrast to the significant preaveraging corrections estimated in the simulations by Zimm13 (about a 10% deviation). Actually, preaveraging should make no contribution in the calculations to first order in e given by Oono and co-w~rkers?~ and the preaveraging effect first arises in a second-order calculation given first by Wang et al.19 for the Kirkwood-Riseman model. The secondorder calculationsindicate a leading preaveraging correction of the hydrodynamic radius of about 20% so that the analytical theory is only qualitatively in agreement with the simulation data. (See Wang et al.l9 for discussion.) The calculations of Oono and c o - w ~ r k e r also s ~ ~differ from ours on account of technical procedures relating to the t expansion of prefactors. Oono and co-workers reexponentiate prefactors while the calculations of refs 6 and 7 do not follow this procedure. The different procedures are illustrated by considering the trivial calculation of the



mean-square end-bend distance of a Gaussian chain. Oono and co-workerssg use units where (d/Z) = 1, d is the dimension of space and 1 is the Kuhn length, so that (R2) of a Gaussian chain is equal

(R2)o= d N 4 ( 1 - e / 4 ) N (D.5a) where N is the chain length. Oono and co-workers exponentiate 1 - t / 4 to order c as

-

+ 0(e2)

(D.5b)

(R2), = 4Ne-'l4 + O(t2)

(D.5c)

1 - e/4

to obtain Reexponentiation as in (D.5b) was followed in the early work by Douglas and Freed,6a but this procedure was strictly avoided in later work6v7when it was realized that errors could be accumulated after many manipulations of this type. Calculations by Oono and c o - ~ o r k e r employ s~~ the reexponentiation procedure, and, as the trivial example (D.5) shows, there can be a significant difference between calculations while at the same time being consistent with t perturbation theory. Finally, we mention that the dynamical renormalization group calculations of Oono and co-workers59 reach the disturbing conclusion that the hydrodynamic properties of Gaussian chains are independent of the hydrodynamic interaction. While 8 polymers are observed to conform reasonably to the nondraining limit,20the application of either the Kirkwood-Riseman or Rouse-Zimm models to Gaussian chains mathematically admits of a crossover from the free-draining to the nondraining l i m i t ~ . ' ~ J ~ ~ ~ O

References and Notes Stockmayer, W. H.; Fixman, M. Ann. N.Y. Acad. Sci. 1953,57, 334. The reptation model takes the cooperative motion of the polymer as a whole to be strongly affected by branching architecture. See: (a) de Gennes, P.-G. J. Chem. Phys. 1971,55572;J . Phys. (Paris) 1975,36,1199. (b) Edwards, S. F.; Doi, M. The Theory of Polymer Dynamics; Clarendon: Oxford, 1988. (c) Pearson, 1983,18, 1. D. S.; Helfand, E. Faraday Symp. Chem. SOC. (a) Roovers, J. Macromolecules 1985,18,1359. (b) McKenna, G. B.; Hadziioannou, G.; Lutz, P.; Hild, G.; Strazielle, C.; Straupe, C.; Rempp, P.; Kovacs, A. J. Macromolecules 1987,20,498. (c) McKenna, G. B.; Plazek, D. J. Polym. Commun. 1986,27,304. (a) Kraus, G.; Gruver, J. T. J. Polym. Sci. A 1965,3, 105. (b) White, J. L. Rubber Chem. Technol. 1969, 42, 257. (c) Haws, J. R.; Wright, R. F. In Handbook of Thermoplastic Elastomers; Walker, B. M., Ed.; Reinhold: New York, 1979. (a) Hadjichristidis, N.; Roovers, J. J . Polym. Sci., Poly. Phys. Ed. 1974, 12, 2521. (b) Roovers, J. Polymer 1979,20, 843. (c) Dodgson, K.; Semylen, J. A. Polymer 1977, 18, 1265. (d) Bezwada, R. S.; Stivala, S. S. Polym. Commun. 1985, 26, 92. Douglas, J. F.; Freed, K. F. Macromolecules 1984,17,2344; 1985, 18, 201. See ref 7, Note 27, Appendix D, and also: (a) Douglas, J. F.; Freed, K. F. Macromolecules 1984,17,1854. (b) Miyake, A.; Freed, K. F. Macromolecules 1983, 16, 1228. (c) Vlahos, C. H.; Kosmas, M. K. Macromolecules 1983,16,1228. (d) Prentis, J. J. J. Phys. A 194, 17, 1723. (e) Ohno, K.; Binder, K. J. Phys. (Les Ulis, Fr.) 1988,49, 1329. Although it is clear on the basis of simple diagrammatic analysis that linear, ring, and star (finite number of arms) polymers belong to the same universality class in the restricted sense that the RG fixed point and exponent u are invariant to changes of branching architecture, the situation is not so evident for many-armed combs and more complicated "branched" species. Ohno and Binder have clarified the factors that determine the universality classes of branched polymers and have calculated the y exponent for a wide class of branching architectures. They also evaluate y for regular stars to second order in e , which is shown in ref 8 to be an improvement over the first-order t calculation of Miyake and Freed.eb A change of the universality class of manyarm (f m ) and fmite-length stars is to be anticipated, however. General diagrammatic arguments suggest that u = 1 / z for selfavoiding stars in the limit f m since only the interarm

-

-

-

interactions dominate. It should be possible to develop a l/f expansion about the f m limit in a fashion parallel to the l / n expansion in the O(n) model (see: Amit, D. J. Field Theory, the Renormalization Group and Critical Phenomena; Scientific: Singapore, 1984). (7) Freed, K. F. Renormalization Group Theory of Macromolecules; Wiley-Interscience: New York, 1987. (8) Kremer, K.; Binder, K. Monte Carlo Simulation of Lattice Models f o r Macromolecules, preprint. (9) (a) Zimm, B. H.; Stockmayer, W. H. J. Chem. Phys. 1949,17, 1301. (b) Casassa, E. F.; Berry, G. C. J . Polym. Sci., Polym. Phvs. 1966. 4. 891. .(c) Berrv. G. C.: Orofino. T. A. J. Chem. Ph;s. 1964; 46, 1614.' ' (10) . , Yamakawa. H. Modern Theorv of Polvmer Solutions: HarDer and Row: New York, 1971. See 'section 36C of Chapter 6'for a summary of estimates for the hydrodynamic ratios calculated from the Rouse-Zimm and Kirkwood-Riseman theories in the preaveraging approximation. (11) Bywater, S. Adu. Polym. Sci. 1979,30,90. For a discussion of arm polydispersity,see also: Burchard, W. Adu. Polym. Sci. 1983, 48, 1. (12) Mazur, J.; McCrackin, F. Macromolecules 1977,10,326;1981, 14, 1214. See also ref 27. J. Batoulis and K. Kremer (Europhys. Lett. 1988, 7, 683) have recently extended the earlier calculations by Mazur and McCrackin to longer chains to find similar results. This work obtains the estimate p,,= 0.28 for 12-arm stars on a fcc lattice. (13) Zimm, B. H. Macromolecules 1984,17,2441. See also: Zimm, B. H. Macromolecules 1980,13,592. (14) Freire, J. J.; Pla, J.; Rey, A.; Plats, R. Macromolecules 1986, 19, 452. (15) (a) Whittington, S. G.; Lipson, J. E. G.; Wilkinson, M. K.; Gaunt, D. S. Macromolecules 1986,19,1241. (b) Lipson, J. E. G.; Gaunt, D. S.; Wilkinson, M. K. Whittington, S. G. Macromolecules 1987, 20, 186. (16) Barrett, A. J.; Tremain, D. L. Macromolecules 1987,20,1687. (17) Daoud, M.; Cotton, J. P. J. Phys. (Paris) 1982,43,531. See also: Birshtein, T. M.; Zhulina, E. B. Polymer 1984,25,1453. For a critical discussion of the "scaling theory" model assumptions, see: Croxton, C. A. Macromolecules 1988,21, 2269. (18) Cherayil, B. J.; Douglas, J. F.; Freed, K. F. J. Chem. Phys. 1987, 87, 3089. See Table I1 of this reference. (19) Wang, S. Q.; Douglas, J. F.; Freed, K. F. J . Chem. Phys. 1986, 85, 3674; 1987,87, 1346. (20) Wang, S. Q.;Douglas, J. F.; Freed, K. F. Macromolecules 1985, 18, 2464. (21) Mansfield, M. L.; Stockmayer, W. H. Macromolecules 1980,13, 1713. (22) Berry, G. C. J. Polym. Sci. A2 1971,9,687. See also ref 5a for samples S141 and S191. (23) Roovers, J.; Bywater, S. Macromolecules 1974, 7, 443. (24) Roovers, J.; Hadjichristidis, N.; Fetters, L. J. Macromolecules 1983, 16, 214. (25) Roovers, J.; Toporowski, P. M. Macromolecules 1981,14,1174. (26) Roovers, J. J . Polym. Sci., Polym. Phys. Ed. 1985,23, 1117. (27) The calculations of Douglas and Freed and Miyake and Freed in ref 6 are based on the same theoretical framework of the RG theory, but Douglas and Freed utilize the exact perturbation theory of the conventional tweparameter perturbation expansion for excluded volume in three dimensions, while the Miyake and Freed calculation involves an additional t expansion of the perturbation theory coefficient. The Douglas and Freed calculation leads to superior results in comparison with experiments and simulation data. The motivation for the method is reviewed by Freed.7 (28) Edwards, S. F.; Singh, P. J. Chem. SOC. Faraday Trans. 2 1979, 75,1001. See also: Muthukumar, M.; Edwards, S. F. J. Chem. Phys. 1983, 78, 2720; Reference 2b, pp 29-31. (29) Guttman, C.; DiMarzio, E. J . Phys. Chem. 1989, 93, 7005. (30) (a) Candau, F.; Rempp, R.; Benoit, H. Macromolecules 1972, 5,627. (b) Zilliox, J. Makromol. Chem. 1972,156,121. (c) Bauer, B. J.; Hadjichristidis, N.; Fetters, L. J.; Roovers, J. E. L., J.. Am. Chem. SOC. 1980,102,2410. (d) Roovers, J.; Toporowski, P. M. Macromolecules 1983, 16, 843. (31) Kolinski, A,; Sikorski, A. J . Polym. Sci., Polym. Chem. 1982, 20,3147. Batoulis and Kremer in ref 12 claim that the 8 point of linear and star polymers are the same for infinite chains and finite f . See ref 35 also. (32) Khasat, N.; Pennini, R. W.; Hadjichristidis, N.; Fetters, L. J. Macromolecules 1988, 12, 1100. (33) Bauer, B. J.; Fetters, L. J.; Graessley, W. W.; Hadjichristidis, N.; Quack, G. Macromolecules 1989,22,2237. (34) Huber, K.; Burchard, W.; Bantle, S.; Fetters, L. J. Polymer 1987, 28, 1990. -

I

4180 Douglas et al. (35) 8 conditions are usually defined in the Monte Carlo simulations by the temperature where the radius of gyration scales with the chain length to the first power. For infinite chains this should coincide's with the 8 condition defined by the vanishing of the second virial coefficient. Mazur and McCrackid2 do not attempt to distinguish between the linear and star species in their cubic lattice calculations, while Kolinsky and Sikorsky31 distinguish between the 8 points of linear and star polymers in their simulations on a diamond lattice. (36) (a) Kosmas, M. K.; Douglas, J. F. J . Phys. A 1988,21, L155. (b) Douglas, J. F.; Kosmas, M. K. Macromolecules 1989,22,2412. (37) Noda, 1.; Horikawa, T.; Kato, T.; Fujimoto, T.; Nagasawa, M. Macromolecules 1970, 3, 795. (38) (a) Edwards, C. J. C.; Stepto, R. F. T.; Semlyen, J. A. Polymer 1982, 23, 865. (b) Higgins, J.; Dodgson, K.; Semlyen, J. A. Polymer 1979,20,553. (c) Dodgson, K.; Semlyen, J. A. Polymer 1977,18,1365. (d) Lutz, P.; McKenna, G . B.; Rempp, P.; Strazielle, C. Makromol. Chem., Rapid Commun. 1986, 7,599. The experimental data for ring hydrodynamic gQ ratios under 8 conditions seem to agree remarkably well with preaveraging hydrodynamic theory. It would seem that if the linear chain hydrodynamic dimensions are affected by preaveraging by an amount on the order 10% due to preaveraging, the apparent success of the preaveraging theory could only arise from the ring hydrodynamic dimensions being altered by preaveraging to nearly the same degree as the linear chain. This possibility of a cancellation of errors in ratios of hydrodynamic properties can be checked by measuring the ratio of the hydrodynamic radius to the radius of gyration under 8 conditions for both the rings and linear polymers. The preaveraging error cannot cancel in such a ratio. Recent data (Hadziioannou, G.; Cotts, P. M.; ten Brinke, G.; Han, C. C.; Lutz, P.; Strazielle, C.; Rempp, P.; Kovacs, A. J . Macromolecules 1987, 20, 493) demonstrates that RH/( S2)1/2does not agree with the preaveraging theory for both polystyrene rings and linear chains in cyclohexane. In summary, the observed data for the hydrodynamic properties of rings seem to be consistent with having preaveraging corrections of a similar magnitude to those in linear polymers and producing a somewhat misleading agreement between the preaveraging hydrodynamic theory predictions for g, and gH with experiment because of a cancellation of errors in such ratios. (39) Roovers, J.; Bywater, S. Macromolecules 1972,5, 384. (40) Bauer, B. J.; Fetters, L. J.; Graessley, W. W.; Hadjichristidis, N.; Quack, G . Macromolecules 1989,22, 2337.

Macromolecules,

Vol. 23, No. 18, 1990

(41) Roovers, J.;Toporowski, P. M. J.Polym. Sci., Polym. Phys. Ed. 1980, I S , 1907. (42) Roovers, J.; Martin, J. J. Polym. Sci., Polym. Phys. Ed., in press. See also: Roovers. J.: Towrowski. P.: Martin. J. Macromolecules iSS9, 22,1897. (43) Duval, M.; Lutz, P.; Strazielle, C. Makromol Chem., Rapid Commun. 1985,6, 71. (44) Mays, J. W.; Hadjichristidis, N.; Fetters, L. J. Polymer 1988, 29, 681. See also ref 11. (45) Hadiichristidis. N.; Roovers. J. Polymer 1985, 26, 1087. (46) RooGers, J. Polymer 1985, 26, 1091: (47) Geiser, D.; Hocker, H. Macromolecules 1980, 13, 653. (48) Roovers, J.; Toporowski, P. M. J . Polym. Sci., Poly. Phys. Ed. 1988,26, 1251. (49) Freire, J. J.; Rey, A.; de la Torre, J. G . Macromolecules 1986, 19, 457; 1987,20, 342. (50) Wilkinson, M. K.; Gaunt, D. S.; Lipson, J. E. G.; Whittington, S. G. Macromolecules 1988,21, 1818. (51) Roovers, J. In Encyclopedia of Polymer Science and Engineering; Wiley: New York, 1985; Vol2, p 478. See also refs 24 and 41. (52) Freed, K. F.; Wang, S. Q.; Roovers, J.; Douglas, J. F. Macromolecules 1988, 21, 2219. (53) Douglas, J. F.; Freed, K. F. J . Phys. Chem. 1984,88, 6613. (54) Roovers, J. Polym. J . 1986, 18, 153. (55) Huber, K.; Burchard, W.; Fetters, L. J. Macromolecules 1984, 17, 541. (56) Roovers, J. Macromolecules 1987,20, 148. (57) Toporowski, P. M.; Roovers, J. J . Polym. Sci., Polym. Chem. Ed. 1986.24, 3009. (58) For attempts to treat knotting in rings, see: (a) ten Brinke, G.; Hadziioannou, G . Macromolecules 1987,20,480. (b) Tanaka, F. J. Chem. Phys. 1987, 87, 4201. (c) Vologodskii, A. V.; Lukashin, A. V.; Frank-Kamanetskii, M. D.; Anshelvich, V. V. Sou. Phys.-JETP (Engl. Transl.) 1974,39, 1059. (59) (a) Oono, Y.; Kohmoto, M. J. Phys. Rev. Lett. 1982,49, 1397; J. Chem. Phys. 1983, 78,520; 79,4620. (60) Oono, Y.; Freed, K. F. J . Phys. A 1982,15, 1931. (61) Des Cloizeaux, J. J. Phys. (Paris) 1981, 42, 635. See also ref 6.

Registry No. Polystyrene, 9003-53-6; polybutadiene, 900317-2; polyisoprene, 9003-31-0.

Characterization of branching architecture through "universal" ratios of

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